Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#38515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5424 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(12,15)(16,18)(19,20,24,28)(21,26,22,25)(23,27,31,34)(29,33,30,32);(4,7)(8,10)(12,15)(13,17)(19,21,24,22)(20,25,28,26)(23,30,31,29)(27,33,34,32);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,22,24,21)(20,26,28,25)(23,30,31,29)(27,33,34,32);(3,5,6,9)(4,8,7,10)(19,20)(21,25)(22,26)(23,27)(24,28)(29,32)(30,33)(31,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,23)(20,27)(21,29)(22,30)(24,31)(25,32)(26,33)(28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7543201);